A338291 Matrix inverse of the rascal triangle (A077028), read across rows.

1, -1, 1, 1, -2, 1, -1, 3, -3, 1, 2, -6, 7, -4, 1, -6, 18, -21, 13, -5, 1, 24, -72, 84, -52, 21, -6, 1, -120, 360, -420, 260, -105, 31, -7, 1, 720, -2160, 2520, -1560, 630, -186, 43, -8, 1, -5040, 15120, -17640, 10920, -4410, 1302, -301, 57, -9, 1

Offset: 0

Author

Werner Schulte, Oct 20 2020

Keywords

• sign
• easy
• tabl

Comments

The columns of this triangle are related to factorial numbers (A000142).

There is a family of triangles T(m;n,k) = 1 + m*k*(n-k) for some fixed integer m (for m >= 0 see A296180, Comments) and 0 <= k <= n. They satisfy the equation T(-m;n,k) = 2 - T(m;n,k). The corresponding matrices inverse M = T^(-1) are given by: M(m;n,n) = 1 for n >= 0, and M(m;n,n-1) = m*(1-n) - 1 for n > 0, and M(m;n,k) = (-1)^(n-k) * m * (m * k*(k+1) + 1) * Product_{i=k+1..n-2} (m*(i+1) - 1) for 0 <= k <= n-2. For special cases of the M(m;n,k) see A338817 (m=-1), and A167374 (m=0), and this triangle (m=1).

Examples

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :      0      1       2      3      4     5     6   7   8  9
================================================================
  0 :      1
  1 :     -1      1
  2 :      1     -2       1
  3 :     -1      3      -3      1
  4 :      2     -6       7     -4      1
  5 :     -6     18     -21     13     -5     1
  6 :     24    -72      84    -52     21    -6     1
  7 :   -120    360    -420    260   -105    31    -7   1
  8 :    720  -2160    2520  -1560    630  -186    43  -8   1
  9 :  -5040  15120  -17640  10920  -4410  1302  -301  57  -9  1
etc.
		

Links

• Werner Schulte, Proof of the Generalized Formula

Crossrefs

Cf. A000007, A000142, A077028.

Programs

PARI

for(n=0,10,for(k=0,n,if(k==n,print(" 1"),if(k==n-1,print1(-n,", "),print1((-1)^(n-k)*(k^2+k+1)*(n-2)!/k!,", ")))))

PARI

1/matrix(10, 10, n, k, n--; k--; if (n>=k, k*(n-k) + 1)) \\ Michel Marcus, Nov 11 2020

Formula

T(n,n) = 1 for n >= 0, and T(n,n-1) = -n for n > 0, and T(n,n-2) = n^2 - 3*n + 3 for n > 1, and T(n,k) = (-1)^(n-k) * (k^2 + k + 1) * (n-2)! / k! for 0 <= k <= n-2.

T(n,k) = (2-n) * T(n-1,k) for 0 <= k < n-2.

T(n,k) = T(k+2,k) * (-1)^(n-k) * (n-2)! / k! for 0 <= k <= n-2.

Row sums are A000007(n) for n >= 0.